密度矩阵的多态几何与整流求和规则
Multi-State Geometry of Density Matrices and Rectification Sum Rules
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中文总结 AI 辅助
本研究发展密度矩阵的多态几何理论，推导适用于多体系统的二阶整流求和规则，经数值验证几何贡献可主导积分响应，低温整流测量可探测绝缘体多态几何。
中文摘要 AI 辅助
量子态的几何已成为理解量子材料线性与非线性响应的关键要素。然而迄今为止，几何与非线性响应的关联仅在零温下干净的无相互作用系统中得到了最充分的理解。在本研究中，我们发展了密度矩阵的多态几何理论，并利用该理论推导了二阶整流的求和规则，且未对无序强度或相互作用强度做任何假设。我们首先证明，热密度矩阵的微扰理论会产生两个对偶的信息论关联与一个近复结构。我们引入了经典信息论中Amari-Chentsov张量的复数量子推广形式——cQAC张量，该张量可捕捉受扰密度矩阵的多态几何。我们推导了绝缘体的频率积分直流整流响应的零温求和规则，其表达式为基态三阶累积量与复畸变张量之差，其中复畸变张量是由cQAC张量构建的多态几何量。这一结果将已知的针对位移电流和非线性霍尔电流的单粒子求和规则推广到了多体系统与一般微扰的情形。针对位移电流，我们将多带绝缘体的几何贡献分解为类粒子项与类空穴项。我们在广义Kane-Mele模型中对该求和规则进行了数值验证，发现几何贡献可主导积分响应。最后，我们证明，尽管该求和规则分解为累积量与几何贡献的形式在非零温下不再成立，但绝缘体的实测求和规则与其零温形式的偏差为由能隙指数小的修正项带来，这使得低温整流测量可用于探测绝缘体的多态几何。
英文摘要
The geometry of quantum states has emerged as a key ingredient in understanding the linear and nonlinear responses of quantum materials. To date, however, the connection between geometry and nonlinear response is best understood for clean, noninteracting systems at zero temperature. In this work, we develop a theory of multi-state geometry for density matrices and use it to derive sum rules for second-order rectification, making no assumptions about the strength of disorder or interactions. We first show that perturbation theory for thermal density matrices gives rise to two dual information-theoretic connections and an almost complex structure. We introduce a complex, quantum generalization of the Amari-Chentsov tensor of classical information theory, the cQAC tensor, which captures the multi-state geometry of the perturbed density matrix. We derive a zero-temperature sum rule for the frequency-integrated DC rectification response of an insulator as a difference between a ground state third cumulant and the complex distortion tensor, a multi-state geometric quantity built from the cQAC tensor. This generalizes known single-particle sum rules for the shift and nonlinear Hall currents to many-body systems and general perturbations. Specializing to the shift current, we resolve the geometric contribution for multiband insulators into particle-like and hole-like terms. We verify the sum rule numerically in a generalized Kane-Mele model, finding that the geometric contribution can dominate the integrated response. Finally, we show that although the splitting of the sum rule into cumulant and geometric contributions does not survive at nonzero temperature, the measured sum rule for insulators differs from its zero-temperature form by corrections exponentially small in the gap, allowing low-temperature rectification measurements to probe the multi-state geometry of insulators.
